I have just built a couple of Aussie Amps & have put 12vdc Triggers in to turn the second amp & MiniDSP on & off. My problem is on turn off, the MiniDSP is badly behaved so it needs to turn off after the amps. I have tried putting a capacitor on the DSP relay to keep it on for a short time, I wired it across the + & - leads to the relay, which worked but it introduced a hum to the system. I was wondering can I wire the capacitor as shown below.

rather than put a delay inside the DSP.
Install two 12V triggers. One on instant off and the other on delayed after loss of power.
That gives you the choice of which trigger to send to any components that can use the facility.

However, a DSP that misbehaves on loss of power is a different problem. Can the misbehaviour cause potential downstream damage? Or severe audio interference?

In this case a mute function on the output of the DSP or at the inputs of the following audio processing equipment would work to "hide" the fault. The mute should activate instantly on loss of power. It can also be triggered by other fault conditions., if you require, eg. on detection of output DC trigger and latch input mute.